| L(s) = 1 | − 2.68e3·9-s + 5.04e3·11-s + 8.24e3·19-s − 1.99e5·29-s + 8.09e4·31-s + 2.82e5·41-s + 1.43e6·49-s − 1.93e6·59-s + 3.77e6·61-s + 5.09e6·71-s + 8.07e6·79-s + 2.41e6·81-s + 1.29e7·89-s − 1.35e7·99-s + 1.94e7·101-s − 1.91e7·109-s − 1.98e7·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 9.33e6·169-s + ⋯ |
| L(s) = 1 | − 1.22·9-s + 1.14·11-s + 0.275·19-s − 1.51·29-s + 0.488·31-s + 0.640·41-s + 1.74·49-s − 1.22·59-s + 2.12·61-s + 1.68·71-s + 1.84·79-s + 0.503·81-s + 1.94·89-s − 1.40·99-s + 1.87·101-s − 1.41·109-s − 1.01·121-s + 0.148·169-s − 0.338·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.389847037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.389847037\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 298 p^{2} T^{2} + p^{14} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 1439150 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2524 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 9331750 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 696354846 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4124 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 134107278 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 99798 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 40480 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 13932162902 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 141402 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 66946399030 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 548078591902 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 937974602250 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 966028 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 1887670 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 3325050217222 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2548232 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 19271301571918 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4038064 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 25265648115558 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6473046 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 124803148841662 T^{2} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.41085260941103423360285204617, −9.673627988316734660092354038891, −9.425173843014377251723223847701, −8.849985823797735572908367336730, −8.741221845375638339307832832095, −7.915202962680560354076133301440, −7.68844884530101600791281481349, −7.05065493164740279683049047723, −6.39137175702943625672075632077, −6.24228042039588999888915254225, −5.39540674156613331853013350269, −5.31879816168852973316723345619, −4.46425353841342197137473091586, −3.69127342864185342393625339971, −3.67322154241250741691821710517, −2.74493776405563246547295244909, −2.27363923884452918363110081256, −1.64765913618588353887328842344, −0.801130031563294673593534160395, −0.49594915105530604032631254653,
0.49594915105530604032631254653, 0.801130031563294673593534160395, 1.64765913618588353887328842344, 2.27363923884452918363110081256, 2.74493776405563246547295244909, 3.67322154241250741691821710517, 3.69127342864185342393625339971, 4.46425353841342197137473091586, 5.31879816168852973316723345619, 5.39540674156613331853013350269, 6.24228042039588999888915254225, 6.39137175702943625672075632077, 7.05065493164740279683049047723, 7.68844884530101600791281481349, 7.915202962680560354076133301440, 8.741221845375638339307832832095, 8.849985823797735572908367336730, 9.425173843014377251723223847701, 9.673627988316734660092354038891, 10.41085260941103423360285204617
